Vol. 7, No. 1, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 4, 541–720
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
Cover Page
Editorial Board
Editors’ Addresses
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Subscriptions
Editorial Login
Author Index
Coming Soon
Contacts
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Comparing a series to an integral

Leon Siegel

Vol. 7 (2014), No. 1, 57–65
Abstract

We consider the difference between the definite integral 0uxeu du, where x is a real parameter, and the approximating sum k=1kxek. We use properties of Bernoulli numbers to show that this difference is unbounded and has infinitely many zeros. We also conjecture that the sign of the difference at any positive integer n is determined by the sign of cos((n + 1)arctan(2π)).

Keywords
polylogarithms, gamma function, Bernoulli numbers
Mathematical Subject Classification 2010
Primary: 33B15
Milestones
Received: 17 July 2012
Revised: 25 May 2013
Accepted: 25 May 2013
Published: 24 October 2013

Communicated by Andrew Granville
Authors
Leon Siegel
Christian-Albrechts-Universität zu Kiel
Christian-Albrechts-Platz 4
24118 Kiel
Germany